# Research: Number Theory

## SERMON 2007

SERMON is a small, friendly, and informal gathering of number theorists and combinatorialists. Faculty members, graduate students, and undergraduate students are all invited to attend, and to give talks if they wish.

The first SERMON took place at UNC Greensboro in 1988. Since then it has taken place every year in at various universities in the southeast: University of Georgia, University of South Carolina, Citadel, College of Charleston, Wake Forest, Virginia Tech, Furman, and Clemson University.

See the home page for the SouthEast Regional Meeting On Numbers for more general information on SERMON and to subscribe to the SERMON e-mail list.

In 2007 SERMON will be held at Wake Forest University in Winston Salem, NC from Friday April 20 to Sunday April 22, 2007. A welcome reception for all participants is held on Friday evening. There will be a party on Saturday night. Talks are held on Saturday and Sunday. The lecture room has overhead projectors as well as computer hookup.

Some funds from the Number Theory Foundation are available to support student participation.

### Registration

Please register by email to sermon@wfu.edu. If you would like to give a talk, please send us a title and an abstract.

### Invited Speakers

Nigel Boston, University of South Carolina

Alice Silverberg, University of California at Irvine

### Abstracts of Talks

#### Nigel Boston: Factoring iterates mod p

We observe and explain patterns in the form of the factorizations of iterates of a given polynomial modulo a prime. Using these and other tools we describe their Galois groups which act naturally on certain rooted trees.

#### Matthew Boylan: Non-vanishing of weakly holomorphic modular form coefficients modulo l and applications

Let f(z) be a half-integral weight modular form with integer coefficients a(n) whose poles (if it has any) are supported at the cusps. Fix a prime l. In this talk, we estimate the number of a(n)’s not divisible by l and give applications to the study of the ordinary partition function, p(n), and other functions of arithmetic interest whose generating functions are of this type. (Joint work with Scott Ahlgren, Univ. of Illinois).

#### Greg Dresden: Look, there’s more to say about Conway’s Look-and-say sequence

We start with a simple game: write down some numbers (like 1,2,2,3) and then write down the description. In this case, the description would be: one 1, two 2’s, and one 3, or in more compact terms, 1,1,2,2,1,3. Now, repeat. What do we end up with? What if we do this in base-2 or base-3? What are some possible variations? This topic was first covered in depth by John Conway, but we have some new results to announce.

#### Jon Grantham: An Unconditional Improvement to the Running Time of the Quadratic Frobenius Test

Damgard and Frandsen have recently demonstrated an extension to the Quadratic Frobenius Test (a probably primality proving test) which has a reduced running time. This improvement can be extended to other versions of the test, but in general depends on the Extended Riemann Hypothesis. I will show how to achieve a speedup intermediate between the original and the new running times that does not depend on any unproven hypotheses.

#### Robert Juricevic: Explicit upper bounds for ∏_{p≤pω(n)} p/(p-1)

Let n≥3 be an integer, σ(n) be the sum of divisors function, and let g(n)=σ(n)/(n log(log n)). Robin (1984) proved that the Riemann Hypothesis is true if and only if g(n)<e^{γ} for n≥ 5041, where γ is Euler’s constant. A consequence of Robin’s theorem is that under the assumption of the Riemann Hypothesis g(n)≤ g(180)=(1.0338…)e^{γ} for n≥121. We describe an algorithm that for a give 0<ε<1 determines all of the exceptions to the inequality ∏_{p≤pω(n)}p/(p-1)<e^{γ}(1+ε)log(log n). By employing this algorithm, we prove unconditionally that σ(n)/(n log(log n)) ≤ σ(180)/(180 log(log 180)) = (1.0338…)e^{γ}, for n≥ 121. This is joint work with Akbary and Friggstad, to appear in Contributions to Discrete Mathematics.

#### Michael Mossinghoff: Sign changes in sums of the Liouville function

Let λ(k) denote the Liouville lambda function, the completely multiplicative function defined by λ(p)=-1 for every prime p. In 1919, Polya noted that the Riemann hypothesis follows if the sum L(n)= ∑_{1≤k≤n}λ(k) does not change sign for large n, and in 1948 Turan noted a similar property for the function T(n)= ∑_{1≤k≤n}λ(k)/k In 1958, Haselgrove proved that both L(n) and T(n) change sign infinitely often, without determining any precise values where a sign change occurs. In 1960, Lehman found an integer n_{0}>1 where L(n_{0})>0, but no specific integer n_{1} had been found where T(n_{1})<0. We describe a recent large computation that has determined the smallest such integer.

#### David Penniston: Arithmetic properties of Maass forms arising from theta series

We investigate the arithmetic properties of the Fourier coefficients of a certain class of Maass forms. As an application we obtain divisibility and distribution results on the coefficients of one of Ramanujan’s mock theta functions.

#### Filip Saidak: Sharper Selberg’s Lemma

We give an explicit connection between the remainder term in the prime number theorem and the error term in Selberg’s prime number lemma, and we discuss some of its consequences.

#### Alice Silverberg: Point counting on CM elliptic curves

In joint work with Karl Rubin, we generalize to the case of arbitrary imaginary quadratic fields earlier results of Gross and Stark on counting the number of points on elliptic curves with complex multiplication, and on finding models for Q-curves. As an application, we give an easy way to distinguish between the twists of an ordinary elliptic curve E over F_{p} in order to identify one with p +1-2U points, when p=U^{2}+dV^{2} with half-integers U and V and E is constructed using the CM method.

#### Ethan C. Smith: Elliptic Curves, Modular Forms, and Sums of Hurwitz Class Numbers

Let H(N) denote the Hurwitz class number. It is known that if p is a prime, then ∑_{|r|<2p1/2} H(4p-r^{2}) = 2p. In this talk, we will investigate the behavior of this sum with the additional condition r=c mod m. Three different methods will be explored for determining the values of such sums. The key ingredients involved are

- counting elliptic curves over F
_{p}, - coefficients of modular forms, and
- the Eichler-Selberg trace formula for Hecke operators.

#### Gang Yu: Finite 2-bases of integers

Given a positive integer N, a set of intgers A is called a 2-basis for N if every n ∈ [0,N] ∩ Z can be represented as n=a+b, a,b ∈ A. In this talk I will give a lower bound for |A| as N → ∞ which improves the existing results.

#### Michael Zieve: Polynomial decomposition

I will discuss the operation of composition on polynomials over a field K, namely f(x) o g(x) = f(g(x)). Every polynomial can be written as a composition of indecomposable polynomials, but this decomposition need not be unique. When K is the complex numbers, Ritt precisely described all sources of nonuniqueness; subsequently this result was extended to other fields of characteristic zero. After reviewing these results, I will present various theorems and examples in positive characteristic, where new phenomena are possible: for instance, there are odd polynomials which decompose into polynomials that are not linear changes of odd polynomials.

I will also present results on the difficult problem of computing the intersection of two subfields of K(x), as well as the reducibility of `variables-separated’ polynomials f(x)-g(y).

### Preliminary Program

All talks will be held on the ground floor of Manchester Hall in room 016.

**Friday, April 20, 2007**

16:00 | Mathematics Colloquium — Filip Saidak: Sharper Selberg’s Lemma |

17:30 | Dinner at the Pit: The Fresh Food Company in Reynolda Hall (Buffet) |

19:00 | Drinks in the Lounge (Manchester Hall room 336) |

**Saturday, April 21, 2007**

10:00 | Nigel Boston: Factoring iterates mod p |

11:00 | Coffee |

11:30 | David Penniston: Arithmetic properties of Maass forms arising from theta series |

12:00 | Michael Zieve: Polynomial decomposition |

12:30 | Lunch |

14:30 | Jon Grantham: An Unconditional Improvement to the Running Time of the Quadratic Frobenius Test |

15:00 | Gang Yu: Finite 2-bases of integers |

15:30 | Coffee |

16:00 | Matthew Boylan: Non-vanishing of weakly holomorphic modular form coefficients modulo l and applications |

16:30 | Ethan C. Smith: Elliptic Curves, Modular Forms, and Sums of Hurwitz Class Numbers |

17:00 | Robert Juricevic: Explicit upper bounds for &prod_{p≤p&omega(n)} p/(p-1) |

19:00 | Reception at Fred Howard’s House |

**Sunday, April 22, 2007**

10:00 | Alice Silverberg: Point counting on CM elliptic curves |

11:00 | Coffee |

11:30 | Greg Dresden: Look, there’s more to say about Conway’s Look-and-say sequence |

12:00 | Michael Mossinghoff: Sign changes in sums of the Liouville function |

### Registered Participants

Ken Berenhaut, Wake Forest

Matthew Boylan, South Carolina

Ezra Brown, Virginia Tech

Nigel Boston, South Carolina

Greg Dresden, Washington and Lee University

Paul Duvall, UNCG

Peter Fletcher

Tim Flowers, Clemson

Jon Grantham

Kevin James, Clemson University

Robert Juricevic, Waterloo

S.B. Karmakar

Hiren Maharaj, Clemson University

David Penniston, Furman University

Ashvin Rajan

Filip Saidak, UNCG

Alice Silverberg

Ethan Smith, Clemson University

David Stone

Christian Sykes, UNCG

Sebastian Pauli, UNCG

Theresa Vaughan

Gang Yu

Michael Zieve